RIES - Find Algebraic Equations, Given Their Solution at MROB

RIES - Find Algebraic Equations, Given Their Solution

ries is a program that takes any number (for example, 2.5063) and produces a list of equations very much like the following:

bash# ries 2.5063 Your target value: T = 2.5063 www.mrob.com/ries 2.x = 5 for X = T - 0.0063 {49} l(x)^2 = S(1) for X = T - 0.00373232 {60} x^2 = 2.p for X = T + 0.000328275 {55} x^x = 1+9 for X = T - 0.000115854 {69} C(C(x)) = l(2) for X = T + 8.77443e-05 {67} x^2+e = 9 for X = T + 3.56063e-05 {63} S(1/x^2) = 1-S(1) for X = T - 3.1577e-05 {82} x-q(e) = C(C(1)) for X = T - 2.55135e-05 {81} x-S(x) = 3v7 for X = T + 2.45325e-05 {88} x.S(2) = S(1)v2 for X = T + 2.05293e-05 {88} x/4+1 = 4v7 for X = T + 6.24679e-06 {91} f.S(x) = C(6) for X = T - 5.93376e-06 {81} x-q(C(1)) = 7L3 for X = T - 3.66369e-06 {94} l(q(x)-1) = n(f/3) for X = T + 1.4647e-06 {97} S(x.f) = 1-l(6) for X = T - 1.14085e-06 {96} x-1/C(2) = S(2)+4 for X = T - 5.34897e-07 {99} S(e^(x-1)) = 1/7^2-1 for X = T + 1.01363e-07 {105} A L B = logarithm to base B of A = ln(A) / ln(B) A v B = A root of B e = base of natural logarithms, 2.71828... p = pi, 3.14159... n = negative f = phi, the golden ratio, (1+sqrt(5))/2 q = square root C(x) = cos(x) l = ln, natural logarithm or log base e S(x) = sin(x) --LHS-- --RHS-- -Total- dead-ends: 201017 246999 448016 CPU time: 0.6 expressions: 16356 19438 35794 distinct: 10872 8836 19708 Memory: 960K Total equations tested: 96064992

Notice the answers are ordered by increasing closeness to your number. It should also be apparent that the simplest equations come first and the more complex ones later on. ries follows the example of continued fractions — as you go to longer equations, you get a closer approximaion to your number, and each approximation is the closest approximation that is available with an equation of that complexity.

ries is highly customizable. You can have it omit symbols (like the symbol for phi, the Golden Ratio) if you don't want it to use them in solutions. You can give it an integer and specify that it limit its search to calculations that come out to be exact integers, and it will figure out the shortest way to construct your number from the digits 1 through 9. ries can find the simplest way to (for example) express the value 27 using only the digit 4 and the four basic operators plus, minus, times and divide.

ries is also very fast. The above run takes less than 2 seconds on a 180-MHz P5 (586 clone) machine. You can specify a longer search to get more (and longer) equations, and it will try over 10¹⁰ equations in less than a minute. On a 733-MHz Pentium-III, ries -l4 tests over 1.4×10¹² equations in 62 seconds and uses 115 megs of memory.

I wrote ries after I was frustrated by services such as Plouffe's Inverterand the Inverse Symbolic Calculator. In case you're wondering, the things that frustrated me are described here.

ries runs under Linux¹, but is is easily ported to nearly any C compiler and OS. The source code is distributed under GPL 2.0. You can also get the manpage source. If you don't already have a copy, you should also retrieve the GNU General Public License version 2.0, which defines the terms under which this source code is made available to you, here.

Stupid Math Tricks

Here are a few cute and/or fun things you can do with ries:

Finding Exact Solutions

You can use ries to iterate Newton's Method. Suppose you know that the cube root of 3 starts with the digits 1.442, and want to find more digits. ries 1.442 yields the result:

x^3 = 3 for X = T + 0.00024957 {51}

Just add a few of the digits it gives:

ries 1.442249
x^3 = 3 for X = T + 5.70307e-07 {51}
ries 1.442249570
x^3 = 3 for X = T + 3.07408e-10 {51}

and so on.

Four Fours

At one time this type of problem was popular: How do you reach various values using four 4's and any number of symbols (for example: 44/44 = 1, 4/4+4/4 = 2, (4+4+4)/4 = 3, etc.). Solutions of this type of problem can be found with a command like:

ries '-S4+-*/' 17 -Ox
x-4*4 = 4/4 (exact match)

which corresponds to the solution 4/4+4×4 = 17.